Digit Properties & Recreational Numbers

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Abstractions about integers defined by digit persistence, representations, sums, permutations, and recreational arithmetic properties.

6 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• 15 puzzle — A four-by-four sliding-tile puzzle whose reachable configurations form one parity class under moves of tiles into a single empty cell.
• Amenable number — A positive integer n admitting a multiset of exactly n integers whose sum and product are both n.
• Dudeney number — A base-dependent natural number that is a perfect cube whose digit sum equals its cube root.
• Keith number — A natural number whose base-b digits seed a k-step Fibonacci-like recurrence that later reproduces the number.
• Persistence of a number — The number of repeated applications of a specified digit operation needed for an integer to reach a fixed point, most commonly a single digit under repeated digit sums or products.
• Taxicab number — The smallest positive integer expressible as a sum of two positive cubes in a specified number of distinct unordered ways.